question
the number of newly reported crime cases in a county in new york state is shown in the
accompanying table, where x represents the number of years since 2011, and y
represents number of new cases. write the linear regression equation that represents
this set of data, rounding all coefficients to the nearest hundredth. using this equation,
estimate the calendar year in which the number of new cases would reach 849.
years since 2011 (x) new cases (y)
0 1105
1 1068
2 1090
3 1052

Question

Accepted Answer

# Explanation:
## Step1: Calculate the mean of $x$ and $y$
Let $x = [0,1,2,3]$, $y=[1105,1068,1090,1052]$
$\bar{x}=\frac{0 + 1+2+3}{4}=\frac{6}{4}=1.5$
$\bar{y}=\frac{1105+1068+1090+1052}{4}=\frac{4315}{4}=1078.75$
## Step2: Calculate the slope $m$
$$
\begin{align*}
m&=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\
(x_{1}-\bar{x})(y_{1}-\bar{y})&=(0 - 1.5)(1105-1078.75)=(-1.5)\times26.25=-39.375\
(x_{2}-\bar{x})(y_{2}-\bar{y})&=(1 - 1.5)(1068 - 1078.75)=(-0.5)\times(-10.75)=5.375\
(x_{3}-\bar{x})(y_{3}-\bar{y})&=(2 - 1.5)(1090 - 1078.75)=(0.5)\times11.25 = 5.625\
(x_{4}-\bar{x})(y_{4}-\bar{y})&=(3 - 1.5)(1052 - 1078.75)=(1.5)\times(-26.75)=-40.125\
\sum_{i = 1}^{4}(x_{i}-\bar{x})(y_{i}-\bar{y})&=-39.375 + 5.375+5.625-40.125=-68.5\
(x_{1}-\bar{x})^{2}&=(0 - 1.5)^{2}=2.25\
(x_{2}-\bar{x})^{2}&=(1 - 1.5)^{2}=0.25\
(x_{3}-\bar{x})^{2}&=(2 - 1.5)^{2}=0.25\
(x_{4}-\bar{x})^{2}&=(3 - 1.5)^{2}=2.25\
\sum_{i = 1}^{4}(x_{i}-\bar{x})^{2}&=2.25+0.25+0.25+2.25 = 5\
m&=\frac{-68.5}{5}=-13.7
\end{align*}
$$
## Step3: Calculate the y - intercept $b$
Using the formula $y=mx + b$, substitute $m=-13.7$, $x = 1.5$, $y = 1078.75$
$1078.75=-13.7\times1.5+b$
$1078.75=-20.55 + b$
$b=1078.75 + 20.55=1099.3$
The linear regression equation is $y=-13.7x + 1099.3$
## Step4: Solve for $x$ when $y = 849$
$849=-13.7x+1099.3$
$13.7x=1099.3 - 849$
$13.7x=250.3$
$x=\frac{250.3}{13.7}\approx18.27$
The year is $2011+18.27\approx2029$

# Answer:
The linear regression equation is $y=-13.7x + 1099.3$ and the year is approximately $2029$

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the mean of \(x\) and \(y\)

Step2: Calculate the slope \(m\)

Step3: Calculate the y - intercept \(b\)

Step4: Solve for \(x\) when \(y = 849\)

Answer: